Generalized inverse

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In mathematics, and in particular, algebra, a generalized inverse of an element x is an element y that has some properties of an inverse element but not necessarily all of them. Generalized inverses can be defined in any mathematical structure that involves associative multiplication, that is, in a semigroup. This article describes generalized inverses of a matrix .

Formally, given a matrix and a matrix , is a generalized inverse of if it satisfies the condition .[1][2][3]

The purpose of constructing a generalized inverse of a matrix is to obtain a matrix that can serve as an inverse in some sense for a wider class of matrices than invertible matrices. A generalized inverse exists for an arbitrary matrix, and when a matrix has a regular inverse, this inverse is its unique generalized inverse.[4]


Consider the linear system

where is an matrix and , the column space of . If is nonsingular, then will be the solution of the system. In this case, C. R. Rao and S. K. Mitra call a regular inverse of .[5] Note that, if is nonsingular, then

Suppose is singular, or . Then we need a right candidate of order such that for all ,


That is, is a solution of the linear system . Equivalently, we need a matrix of order such that

Hence we can define the generalized inverse or g-inverse as follows: Given an matrix , an matrix is said to be a generalized inverse of if [7][8][9]


The Penrose conditions define different generalized inverses for and

where indicates conjugate transpose. If satisfies the first condition, then it is a generalized inverse of . If it satisfies the first two conditions, then it is a reflexive generalized inverse of . If it satisfies all four conditions, then it is a pseudoinverse of .[10][11][12][13] A pseudoinverse is sometimes called the Moore–Penrose inverse, after the pioneering works by E. H. Moore and Roger Penrose.[14][15][16][17][18] When is non-singular, and is unique, but in all other cases, there is an infinite number of matrices that satisfy condition (1). However, the Moore–Penrose inverse is unique.[19]

There are other kinds of generalized inverse:

  • One-sided inverse (right inverse or left inverse)
    • Right inverse: If the matrix has dimensions and , then there exists an matrix called the right inverse of such that , where is the identity matrix.
    • Left inverse: If the matrix has dimensions and , then there exists an matrix called the left inverse of such that , where is the identity matrix.[20]


Reflexive generalized inverse[edit]


Since , is singular and has no regular inverse. However, and satisfy conditions (1) and (2), but not (3) or (4). Hence, is a reflexive generalized inverse of .

One-sided inverse[edit]


Since is not square, has no regular inverse. However, is a right inverse of . The matrix has no left inverse


The following characterizations are easy to verify:

  1. A right inverse of a non-square matrix is given by .[21]
  2. A left inverse of a non-square matrix is given by .[22]
  3. If is a rank factorization, then is a g-inverse of , where is a right inverse of and is left inverse of .
  4. If for any non-singular matrices and , then is a generalized inverse of for arbitrary and .
  5. Let be of rank . Without loss of generality, let
where is the non-singular submatrix of . Then,
is a g-inverse of .


Any generalized inverse can be used to determine whether a system of linear equations has any solutions, and if so to give all of them. If any solutions exist for the n × m linear system


with vector of unknowns and vector of constants, all solutions are given by


parametric on the arbitrary vector , where is any generalized inverse of . Solutions exist if and only if is a solution, that is, if and only if .[23]

See also[edit]


  1. ^ Ben-Israel & Greville (2003, pp. 2,7)
  2. ^ Nakamura (1991, pp. 41–42)
  3. ^ Rao & Mitra (1971, pp. vii,20)
  4. ^ Ben-Israel & Greville (2003, pp. 2,7)
  5. ^ Rao & Mitra (1971, pp. 19–20)
  6. ^ Rao & Mitra (1971, p. 24)
  7. ^ Ben-Israel & Greville (2003, pp. 2,7)
  8. ^ Nakamura (1991, pp. 41–42)
  9. ^ Rao & Mitra (1971, pp. vii,20)
  10. ^ Ben-Israel & Greville (2003, p. 7)
  11. ^ Campbell & Meyer (1991, p. 9)
  12. ^ Nakamura (1991, pp. 41–42)
  13. ^ Rao & Mitra (1971, pp. 20,28,51)
  14. ^ Ben-Israel & Greville (2003, p. 7)
  15. ^ Campbell & Meyer (1991, p. 10)
  16. ^ James (1978, p. 114)
  17. ^ Nakamura (1991, p. 42)
  18. ^ Rao & Mitra (1971, p. 50–51)
  19. ^ James (1978, pp. 113–114)
  20. ^ Rao & Mitra (1971, p. 19)
  21. ^ Rao & Mitra (1971, p. 19)
  22. ^ Rao & Mitra (1971, p. 19)
  23. ^ James (1978, pp. 109–110)


  • Ben-Israel, Adi; Greville, Thomas N.E. (2003). Generalized inverses: Theory and applications (2nd ed.). New York, NY: Springer. ISBN 0-387-00293-6. 
  • Campbell, S. L.; Meyer, Jr., C. D. (1991). Generalized Inverses of Linear Transformations. Dover. ISBN 978-0-486-66693-8. 
  • James, M. (June 1978). "The generalised inverse". Mathematical Gazette. 62: 109–114. doi:10.2307/3617665. 
  • Nakamura, Yoshihiko (1991). Advanced Robotics: Redundancy and Optimization. Addison-Wesley. ISBN 0201151987. 
  • Rao, C. Radhakrishna; Mitra, Sujit Kumar (1971). Generalized Inverse of Matrices and its Applications. New York: John Wiley & Sons. p. 240. ISBN 0-471-70821-6. 
  • Zheng, B; Bapat, R. B. (2004). "Generalized inverse A(2)T,S and a rank equation". Applied Mathematics and Computation. 155: 407–415. doi:10.1016/S0096-3003(03)00786-0.